Answer:
[tex] \cos(2A) = { \cos }^{2} A - { \sin }^{2} A \\ = { \cos }^{2} A - \frac{1}{ \csc {}^{2} A} \\ \\ = \frac{( { \cos}^{2} A. \csc {}^{2}A ) - 1}{ { \csc }^{2} A} \\ \\ = \frac{( \frac{ { \cos}^{2}A }{ { \sin }^{2} A}) - 1 }{ { \csc }^{2} A} \\ \\ = \frac{ { \cot}^{2}A - 1 }{ { \csc}^{2} A} [/tex]
but csc²A = cot²A + 1:
[tex] = \frac{ { \cot}^{2}A - 1 }{ { \cot }^{2}A + 1 } [/tex]
# proved
